Question

If the point P(x,y) is equdistant from point $A(a+b,b-a)\&B(a-b,a+b).Provethatbx-ay$.

Answer 1

$PA=PB\Rightarrow (x-(a+b){)}^2+(y-(b-a){)}^2=(x-(a-b){)}^2+(y-(a+b){)}^2\Rightarrow x^2-2x(a+b)+(a+b{)}^2+y^2-2y(b-a)+(b-a{)}^2=x^2-2x(a-b)+(a-b)(a-b{)}^2+y^2-2y(a+b)+(a+b{)}^2\Rightarrow -2x(a+b)-2y(b-a)=-2x(a-b)-2y(a+b)\Rightarrow -2x(a+b-a+b)-2y(b-a-a-b)=0\Rightarrow -2x\left(2b\right)-2y(-2a)=0\Rightarrow 4bx+4ay=0\Rightarrow 4ay+4bx\Rightarrow ay=bx$